Shortcut to Find Square of a Number

Today I will discuss a very simple method of finding square of numbers between 26 to 74 mentally. In the subsequent post we will cover higher numbers. So keep watching this space to learn squaring any number within your mind

Square (also called perfect square) is an integer that is the square of an integer; in other words, it is the product of some integer with itself. So, for example, 9 is a square number, since it can be written as 3 × 3.

How to find the square of any number?

To apply this method you should know squares of 1 to 25 by heart. You can refer to this table to learn the same.

Number

Square

Number’

Square’

1

1

13

169

2

4

14

196

3

9

15

225

4

16

16

256

5

25

17

289

6

36

18

324

7

49

19

361

8

64

20

400

9

81

21

441

10

100

22

484

11

121

23

529

12

144

24

576

*

*

25

625

For finding square of any number between 26 to 75

Step 1. Find the difference between 50 and the number you want to square.

To find the square of any small number, we can easily do it by using expansions.
For example, if we want to square 47 , we can do it by changing the number into two numbers.
47×47=(50-3)the whole square.
Then,by using expansions, the result is
50square + 3square – 2×50×3
Which is nothing but 2203.
The answer!!!

Make it (178*900)/4. Now 89*9 can be calculated more easily (89 came after cancelling 178/4 by 2 each). If you see closely, 89*9 be calculated as 90*9 and then subtract 9 from there which will come out to be 801. Now divide 80100 by 2 which will be 40050 which again you do not need to CALCULATE but can find out just by looking at numbers. So the whole calculation can be solved without using a pen.

For squaring of any 2 digit number.
Example:-
(67)^2=
Get to its nearest 10’s
1st. As 67, its nearest 10 is 70, so 70-67=3
Then 67-3=64,
64*70=4480—1,
2nd. As 70 is 3 more than 67, so
(3)^2=9—2,
Adding 1&2- 4480+9=4489 s here it is..
(67)^2=4489.

a method to find square of a no.
• Consider number 121 to find square.
• Separate digits as 12 & 1.
• Now do Square of unit digit number i.e. 1’square=1, so our unit digit number is 1 now in this case.
• Then multiply both separated Digits which are earlier separated ie.12 & 1
• =12 X 1=12
• Now double the result of multiplication i.e. Double of 12= 24. And place it left after unit digit number i.e. 2 41.( 2 is Carry)
• Separate carry 2 then
• Now At last square 12 i.e. 12’square=144 and add earlier Carry to it.
• So we get 144+2(carry)=146
• Finally we obtained Square as 14641..
Let us have a look at one more example.
151’square:
• 1’s square=1.
• 2 x (15 x 1) = 3 0 (Double of 15 and 1 and 3 is carry now.)
• Now 15’square=225+earlier carry i.e.3=228
• So Final Answer is 22801
here’s another way
Let us first find square of 11 using formula:
11’square=11+1/1square=12/1=121.
The formula is self explanatory. However, let me explain it in detail for more clarification.
• The slash is used just as a operator.
• Our operating zone is 10 X 1 or simply 10.
• 11 is more than 10.
• We add 1 to 11to make 12.
• The number of digits after the slash can be only one.
• If the number of digits after the slash exceeds one, then we place only the rightmost digit on the extreme right after the slash, and the remaining gets added to the number on the left hand side of the slash.
• Now have a look at few more examples for better understanding.
12’square=12+2/2’square=14/4=144
13’square=13+3/3’square=16/9=169.
14’square=14+4/4’square=18/16 (Apply step no 6 here) 18/16=18+1/6=196.
15’square=15+5/5’square=20/25=20+2/5=225.
16’square=16+6/6’square=22/36=22+3/6=256.
You can work like this up to 19’square.But for 20 formula is slightly change.
The slight Change in formula as follows:
21’square=2 X (21+1)/1’square= 2 X (22)/1=44/1=441.
This change is because now we are operating in the 10 X 2 Zone. Similarly we can calculate Square of 31 but with slight change as follows:
31’Square=3 X (31+1)/!’square= 3 X (32)/1=96/1=961.
By these methods explained you can easily calculate and memorize the squares of numbers up to 99 with out much hassle.
Master Formula to Calculate Square in 10 Seconds.:
Till now we have seen various formulae to calculate square of the number. Now I am giving such a master Formula by which you can orally calculate Square of the number with in 10 Seconds.
11’square=121. 12’square=144
111’square=12321 121’square=14641

For sqaure root:-
(for all non perfect sqaures)
Find two perfect squares that your square falls between. For example if I am trying to find the square root of 12, then I know my number is going to fall between the square root of 9 (3^2=9) and the square root of 16 (4^2=16).
Divide your square by one of these two square roots. Therefore, I am going to divide my square, 12, by one of the square roots 3 or 4. I will choose 3. So, 12/3 = 4.
I will average the result from Step 2 with the root I divided by. So, I take my answer from Step 2 (4), and will average this with the root I chose to divide my square by in Step 2 (3). Therefore, (4+3)/2 = 3.5. The square root of 12 is 3.5!
try this with large numbers. for very small numbers like 2, 3 etc it doesn’t work i.e. as you try to find the square root of larger numbers you will get more precise answer……

I know one easiest way to find the square root.
Ex: I want to find out the square of 76
Explanation :
a) 76 is nearer to 80 than 70
b) square of 80 is 6400 we know that ( 8* 8 = 64*100= 6400)
c)know multiply the 80 with twice difference of 80 and 76 (80-76 =4) (4*2 =8)
d) 8*80 = 640
e) subtract 640 from 6400 and add 6 (it is the last digit of 6 sqare) at ones digit. you got the answer

Exception=
1) if the number ends with 4 we need to add 16 or ends with 5 we need to add 25

For sqaure root:-
(for all non perfect sqaures)
Find two perfect squares that your square falls between. For example if I am trying to find the square root of 12, then I know my number is going to fall between the square root of 9 (3^2=9) and the square root of 16 (4^2=16).

Divide your square by one of these two square roots. Therefore, I am going to divide my square, 12, by one of the square roots 3 or 4. I will choose 3. So, 12/3 = 4.

I will average the result from Step 2 with the root I divided by. So, I take my answer from Step 2 (4), and will average this with the root I chose to divide my square by in Step 2 (3). Therefore, (4+3)/2 = 3.5. The square root of 12 is 3.5!

try this with large numbers. for very small numbers like 2, 3 etc it doesn’t work i.e. as you try to find the square root of larger numbers you will get more precise answer……

HI,
CUBE ROOT OF 512000
STEP1 :- CLOSE THE LAST THREE NUMBERS AND THE REMAINND NUMBERS =512
SEE THE IN WHICH NUMBER IS THERE IF IT IS IN BETWEEN THEN LEAST NUMBER CUBE
STEP2 :- AND YOU SEE THE LAST NUMBER 0 IS 0 AND IF IT IS 4 THEN YOU SEE THE CUBE ROOT OF 4 IS 64 YOU SEE ONLY LAST NMBER

Thanks Gokul for your suggestion. BTW the answer is 1369 and not 1379.
Actually this is an application of the formula: (a+b)^2 = a^2 + 2*a*b + b^2; here a = 30 and b = 7.
Arranging the same thing vertically will make it easier for you to understand –
900
420
49
——-
1369

If written in the manner suggested by Gokula (vedic maths way), it becomes simpler.

I will not say there aren’t simpler methods of finding square than the one mentioned in this post. But this method can be used for large variety of case.

In any case, I appreciate your efforts of describing various methods. I will be happy to post any method suggested by you on QM (if it’s not already published on QM ). I must say your earlier posts were superb.

With due respect to all, I feel, there are shorter methods for finding squares of numbers without need to remember squares of 1 to 25.

Pl. go through the three tricks below:

1. A quick way to square numbers that end in 5:
75^2 = 5625

75^2 means 75 x 75.
The answer is in two parts: 56 and 25. The last part is always 25.
The first part is the first number, 7, multiplied by the number “one more”, which is 8:
so 7 x 8 = 56

Similarly 852 = 7225 because 8 x 9 = 72.

Before trying to square numbers which do not end in 5, please learn the following trick.

2. Method for multiplying numbers where the first figures are the same and the last figures add up to 10.
32 x 38 = 1216
Both numbers here start with 3 and the last figures (2 and 8) add up to 10.
So we just multiply 3 by 4 (the next number up) to get 12 for the first part of the answer.
And we multiply the last figures: 2 x 8 = 16 to get the last part of the answer.

And 81 x 89 = 7209
We put 09 since we need two figures as in all the other examples.

@Nikhil: Ofcourse you can use the same method to find the square of numbers greater than 74. But in my next method I will give you the extension of this technique to find squares of number between 76 to 124 and so on.

Hi , this method seems correct, but if I try to find square of say 89 then from your method i need to know (89-50)= 39^2. so in that case we may have to use to this method twice, at first to find 39^2 and then 89^2